In the following we keep for the parameters in the birth and
death rate the same values as in the case of global
interaction, i.e., \(r_{b0} = 1\) , \(r_{d0} = 0.1\) , \(\alpha =0.02\) , \(\beta =0\) . For these parameter values the number of
bugs fluctuates only weakly around the mean value. Differently
from the case of global interaction, now the average number of
bugs in the system is influenced not only by the birth and
death rates, but also by the diffusion, i.e., in the case of
Brownian bugs by \(\kappa \)  and in the case of Lévy bugs by
\(\kappa _\mu \)  and \(\mu \) , see Fig. REF . The smaller is
\(\kappa \) , \(\kappa _\mu \)  or \(\mu \) , the larger the particle
number. At the same time Figs. REF  and REF 
indicate that by decreasing \(\kappa \)  or \(\kappa _\mu \)  the
linear width of the clusters becomes smaller, the particle
density in the clusters higher, and the density between the
clusters lower (c.f. Sec. REF  and see also
Ref. [1]}). Somehow counterintuitively, the
effect of decreasing \(\mu \)  seems to have the same effects, as
commented above for the noninteracting and global cases.
Furthermore, the value of \(\kappa \)  or \(\kappa _\mu \)  and \(\mu \) 
seems to weakly influence the number of clusters in the system:
In Figs. and REF  and REF  smaller values lead
to larger number of clusters. This observation is not explained
by the linear instability analysis of
[2]}, [1]}.
<FIGURE>